Q: Which of the following statements about the vector (cross) product is correct?

$$\vec{A} \times \vec{B} is $$perpendicular to both $$\vec{A}$$ and $$\vec{B}.$$

Q: Which of the following is the correct formula for the magnitude of the cross product of two vectors $$\vec{A}$$ and $$\vec{B}$$?

$$|\vec{A} \times \vec{B}| = |\vec{A}|\ |\vec{B}|\sin\theta$$

Q: Which of the following is the correct determinant form for the cross product $$\vec{A} \times \vec{B}$$?

$$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$$

Question 1

A point in the Cartesian coordinate system has coordinates $$(-3.50, -2.50) m. $$What are its polar coordinates $$(r, 	heta)$$, with $$	heta$$ measured counterclockwise from the positive x-axis?

Accepted Answer

$$(4.30\ \mathrm{m},\ 216^\circ)$$

Question 2

Which of the following is a correct statement about the difference between scalar and vector quantities?

Accepted Answer

A scalar has only magnitude, while a vector has both magnitude and direction.

Question 3

A hiker walks $$25.0 km $$southeast (i.e., $$45.0^\circ$$ south of east), then $$40.0 km in a $$direction $$60.0^\circ$$ north of east. What is the x-component of her total displacement?

Accepted Answer

$$37.7\ \mathrm{km}$$

Question 4

Given vectors $$\vec{A} = 3\hat{i} + 4\hat{j}$$ and $$\vec{B} = 5\hat{i} - 2\hat{j}$$, what is $$\vec{A} \cdot \vec{B}$$?

Accepted Answer

$$7$$

Question 5

Which of the following best describes the commutative property of vector addition?

Accepted Answer

$$\vec{A} + \vec{B} = \vec{B} + \vec{A}$$

Question 6

If $$\vec{A} = 2\hat{i} + 3\hat{j} - \hat{k}$$ and $$\vec{B} = -\hat{i} + 4\hat{j} + 2\hat{k}$$, what is $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$10\hat{i} - 5\hat{j} + 11\hat{k}$$

Question 7

A vector $$\vec{A}$$ has components $$A_x = -6$$ and $$A_y = 8. $$What is the magnitude and direction (angle from the positive x-axis, counterclockwise) of $$\vec{A}$$?

Accepted Answer

$$10,\ 127^\circ$$

Question 8

Which of the following is NOT a property of unit vectors?

Accepted Answer

They can have any magnitude.

Question 9

If $$\vec{A}$$ and $$\vec{B}$$ are perpendicular, what is their scalar (dot) product?

Accepted Answer

$$0$$

Question 10

A force $$\vec{F} = 10\hat{i} + 5\hat{j} N $$acts on a particle that moves through a displacement $$\vec{d} = 3\hat{i} - 4\hat{j} m. $$What is the work done by the force?

Accepted Answer

$$10 N$$·m

Question 11

Which of the following statements about the vector (cross) product is correct?

Accepted Answer

$$\vec{A} 	imes \vec{B} is $$perpendicular to both $$\vec{A}$$ and $$\vec{B}.$$

Question 12

A vector $$\vec{A}$$ has a magnitude of $$12$$ units and points $$60^\circ$$ above the positive x-axis. What are its $$x$$ and $$y$$ components?

Accepted Answer

$$A_x = 12\cos 60^\circ = 6.0,\ A_y = 12\sin 60^\circ = 10.4$$

Question 13

Which of the following is the correct formula for the magnitude of the cross product of two vectors $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

$$|\vec{A} 	imes \vec{B}| = |\vec{A}|\ |\vec{B}|\sin	heta$$

Question 14

A vector $$\vec{A} is $$multiplied by a negative scalar $$-3. $$What happens to the direction and magnitude of the resulting vector?

Accepted Answer

The magnitude is tripled and the direction is reversed.

Question 15

If $$\vec{A} = 4\hat{i} - 3\hat{j}$$ and $$\vec{B} = -2\hat{i} + 5\hat{j}$$, what is $$\vec{A} + \vec{B}$$?

Accepted Answer

$$2\hat{i} + 2\hat{j}$$

Question 16

Which of the following is a valid operation with vectors?

Accepted Answer

Adding two displacement vectors

Question 17

A vector $$\vec{A}$$ has components $$A_x = -5$$ and $$A_y = 12. $$What is the angle $$	heta (in $$degrees) it makes with the positive x-axis?

Accepted Answer

$$112.6^\circ$$

Question 18

Which of the following is the correct determinant form for the cross product $$\vec{A} 	imes \vec{B}$$?

Accepted Answer

$$\vec{A} 	imes \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}$$

Vectors and Coordinate Systems: Study Notes for College Physics

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